Large intravalley scattering due to pseudo-magnetic fields in crumpled graphene

The pseudo-magnetic field generated by mechanical strain in graphene can have dramatic consequences on the behavior of electrons and holes. Here we show that pseudo-magnetic field fluctuations present in crumpled graphene can induce significant intravalley scattering of charge carriers. We detect this by measuring the confocal Raman spectra of crumpled areas, where we observe an increase of the D'/D peak intensity ratio by up to a factor of 300. We reproduce our observations by numerical calculation of the double resonant Raman spectra and interpret the results as experimental evidence of the phase shift suffered by Dirac charge carriers in the presence of a pseudo-magnetic field. This lifts the restriction on complete intravalley backscattering of Dirac fermions.

The pseudo-magnetic field generated by mechanical strain in graphene can have dramatic consequences on the behavior of electrons and holes. Here we show that pseudo-magnetic field fluctuations present in crumpled graphene can induce significant intravalley scattering of charge carriers. We detect this by measuring the confocal Raman spectra of crumpled areas, where we observe an increase of the D'/D peak intensity ratio by up to a factor of 300. We reproduce our observations by numerical calculation of the double resonant Raman spectra and interpret the results as experimental evidence of the phase shift suffered by Dirac charge carriers in the presence of a pseudo-magnetic field. This lifts the restriction on complete intravalley backscattering of Dirac fermions.
With the discovery of the half integer quantum Hall effect in graphene 1,2 and of topological insulators, 3,4 Berry phase effects have taken center stage in condensed matter research. In graphene and other crystals with a honeycomb structure, charge carriers have a sublattice and valley degree of freedom, described in a continuum Dirac model as a pseudospin 5,6 . As a consequence of this spin-like property, electrons or holes belonging to the two inequivalent valleys (K and K', see Fig. 1) acquire a Berry phase of π and −π respectively during a cyclotron orbit. The importance of pseudospin and the Berry phase are most striking when perturbations are smooth on the atomic scale, ie. sublattice symmetry still holds. In this case, scattering between the two valleys is suppressed and the pseudospin is conserved 5 , leading to some important effects that are the hallmark of graphene, such as weak antilocalization 7 , the half integer quantum Hall effect 1,2 and Klein tunneling 5,8 . Importantly, complete backscattering from a state |−q to |q (see Fig. 1) is forbidden, due to pseudospin conservation 5 (q is the crystal momentum measured from the K point in the Brillouin zone). This was first shown and explained for metallic carbon nanotubes 9,10 and is crucial for the exceptional mobility of graphene 5,11 .
Here we show that scattering on strain fluctuations in graphene can lift the restriction on complete backscattering. This is demonstrated by confocal Raman spectroscopy measurements of crumpled graphene. We measure a giant increase in the D' peak intensity. This Raman peak originates from a resonant Raman process which involves intravalley backscattering of charge carriers. The intravalley to intervalley scattering peak intensity ratio is found to be I D /I D ≈ 30, in contrast to the usual value of ≈0.1 12 . Since the strain induced pseudo-magnetic field (B ps ) couples to the pseudospin 13,14 , the enhancement of the D' peak at 1620 cm −1 is due to the extra phase acquired by charge carriers undergoing Raman scattering on strain fluctuations. Thus, in contrast to a scalar potential, backscattering of Dirac particles is no longer forbidden. We reproduce our measurement results, using numerical calculation of the double resonant Raman processes.
Mechanical deformations in 2D materials with a honeycomb atomic structure naturally give rise to a two component pseudogauge field, which is directly proportional to the strain tensor components [15][16][17][18][19] . These strain induced fields have a scalar (V (r)) and a vector (A(r)) component 18 , being analogous to an electrostatic potential and a magnetic vector potential that has opposite sign in the two valleys. The latter giving rise to a pseudomagnetic field (B ps ) 18 . For graphene supported on hexagonal BN, B ps is especially strong near bubbles (hundreds of Tesla) and has a major influence on transport properties 20 . Furthermore, in the highest mobility heterostructure devices it is very likely that random strain fluctuations are the main factors limiting mobility through intravalley scattering 21 . This type of scattering is characterized by small changes in the charge carrier momentum and dominates if the scattering potentials are smooth on the atomic scale, such as charged impurities [22][23][24] , or strain fluctuations 21 . Such scattering processes are mostly explored in charge transport experiments through weak (anti)localization measurements 7,10,25 . In supported graphene B ps appears due to random strain fluctuations, stemming from non perfect stacking and interaction with the substrate. For SiO 2 supported graphene, B ps has values of the order of 1T and adds an extra phase to the wave function of the scattered carrier, much like a real magnetic field, suppressing the weak anti-localization effect 25 .
Confocal Raman spectroscopy is another powerful tool to measure strain fluctuations in graphene. Mechanical deformation induced softening or hardening of the phonon mode energy is detectable through the shift, splitting and broadening of the G and 2D peaks 21,26,27 . However, until now direct detection of the scattering on strain fluctuations has been lacking. Both small (intravalley) and large momentum (intervalley) scattering is measurable separately via the double resonant D' and D peaks at ≈1620 cm −1 and ≈1350 cm −1 . Lattice defects produce both intra-as well as intervalley scattering, giving contribution to both D' and D peaks, smooth defects which are less efficient at producing large momentum change, mostly contribute to the D' peak. Numerical calculations by Venezuela et al. 28 , show that for closely packed alkali metals on graphene, acting as Coulomb scattering centers, the D' peak is undetectably small. Furthermore, for lattice defects, able to induce both types of scattering, the measured ratio of intensities is I D /I D ≈ 0.1 12 . This is much smaller than the expected value of ∼0.5 from analytical theory of the double resonant processes 29 . If the perturbation doesn't distinguish between the two sublattices, it is unable to flip the pseudospin (σ). Therefore, complete backscattering involving small changes in momentum from |−q to |q is forbidden by pseudospin conservation. (b) Intravalley scattering responsible for the D' Raman band of graphene.
The double resonant D' process is sketched in Fig. 1b. After the creation of the electron/hole pair the largest contribution to the D' intensity is given by processes involving scattering by both an electron and a hole 28,30 . Backscattering of the electron and/or hole involve an elastic defect scattering with small momentum change and a scattering involving an LO phonon along the ΓM direction 28,31 . It has been shown previously that the process involves a small portion of phonon phase space, as well as relatively small regions of the Dirac cone 28,30 . Since the biggest contribution to the intensity involves backscattering within a single valley along ΓM, this process necessarily involves a flip in pseudospin. Indeed it is suspected by Rodriguez-Nieva et al 29 that the pseudospin related phases of the excited electron and hole play a dominant role in suppressing the backscattering necessary for the D' peak. Since Coulomb scatterers do not change the phase of the charge carrier wave functions, the resulting suppression of backscattering straightforwardly explains the immeasurably small 28 intensity of the D' peak. The situation changes drastically if the scattering potential has a vector component, ie. there is a sizable pseudo-magnetic field involved in the defect scattering. If the charge carriers stay within the same valley, this changes the phase of the electron or hole similarly to a real magnetic field, enabling backscattering.
To detect Raman scattering from strain fluctuations with a sizable B ps , we have measured confocal Raman maps on crumpled graphene flakes exfoliated onto a SiO 2 surface. We have used to our advantage that during the exfoliation some flakes tend to crumple (see Fig. 2a). Crumpling enhances B ps caused by the strain fluctuations by at least a factor of 100 compared to the ∼1 Tesla 25 resulting from surface roughness of SiO 2 . Additionally, we found the same results on samples that were crumpled using mild annealing (see Supplement).
AFM measurements show that the graphene layer consists of a network of folds on it's surface. Measuring the Raman spectra with a spatial resolution of ∼500 nm and excitation wavelength of 532 nm, we map the intensity of the D' peak ( Fig. 2c) within the crumpled area marked in Fig. 2a. The most striking feature of the map is that the crumpled regions show increased D' intensity, the region with the strongest enhancement marked by a red circle. To shed more light on the Raman scattering we plot the complete Raman spectrum of this region in Fig. 2d. The most surprising feature of the spectrum is the extremely high D' peak, having 20% the intensity of the G peak. This is unprecedented because the D' peak is only observed when the sample contains strong lattice defects 12,28,32 , and it is always accompanied by the D peak, with its intensity being mostly ∼10% of the D intensity (I D /I D ≈ 0.1) 12,33 . This result has to be considered in the light that in the area where the spectrum was measured there are no graphene edges, therefore we also see a D peak, which is just barely larger than the background (see inset in Fig 2d). Furthermore, the graphene flake shows the pristine Raman spectrum of graphene in the uncrumpled areas. Ratios I D /I D ≈ 10 have been also found in samples where the laser spot contains both crumpled areas and edges (see supplement).
It is known that overlapping graphene layers can produce a non dispersive peak around 1625 cm −1 34-36 . To make sure that the peak around 1620 cm −1 is indeed the D' peak, we measured the change in the peak position with changing excitation wavelength, using: 488 nm, 532 nm and 633 nm. The data for the sample in area discussed here can be seen in Fig. 2e, with the graph color corresponding to the excitation laser color. The measured dispersion is 10 cm −1 /eV and matches the values on two other samples (see supplement), as well as the expected dispersion of the D' peak (see Fig.  2f) 32,33 .
Within the literature there have been some observations of a barely mea- surable D' peak on graphene supported on nanosized pillars 37 and nanoparticles 38 , exhibiting wrinkling. However, it is unclear what the D to D' intensity ratio is in these experiments. In similar experiments, the D' peak is obscured by the presence of the polymer substrate 39 . We reproduce our measured D' peak intensities by numerical calcula- tions of the double resonant processes 28,41 . To explore the pseudo-magnetic field within crumpled graphene and to construct a theoretical model for the Raman scattering potential, we study the wrinkling of graphene, using the LAMMPS molecular dynamics code 40 (for details see Supplement). We consider two model geometries that make up a crumpled graphene sheet 42 . One involves a single fold (Fig. 3b), seen all over the crumpled sample (Fig. 2b), the other involves a double fold (Fig. 3c), constructed by creating a second fold in a singly folded graphene. The double fold shows and increase in strain by a factor of 10 compared to the single fold, as evidenced by the color scale on the atoms in Fig. 3b,c. Computing the pseudo-magnetic field B ps , directly from the modulation of the atomic positions 43 we find a maximum B ps of 20 T in the single fold, and as expected from the difference in strain, the double fold shows a B ps in the 200 T range (Fig. 3e,f). This B ps originates from the modulation of the inter-atom hopping due to strain 15,16,44 . However, in cases where the curvature of the graphene sheet is large, there is another sizable component to the pseudo-magnetic field, due to the hybridization of the σ and π bonds 17 . The total vector potential then is the sum of the hopping induced A(r) and the curvature induced A σπ (r). The experimentally relevant part of the vector potential 13 , the pseudo-magnetic field, is then formed by the rotor of the sum of these two contributions: B ps = ∇ × (A + A σπ ). Using the formula for A σπ from Rainis et al 45 , we calculate the B ps for our fold. For a radius of curvature R of a fold parallel to the zigzag direction, we have A σπ x (r) = 3ε ππ a 2 /8R(r) 2 and A σπ y (r) = 0, where ε ππ ≈ 3 eV 17,45 and a = 1.42 Å. In our calculations R is around 3 Å, corresponding to the 2-3 Å measured by Annett et al. 46 , while Rainis et al 45 calculate 7 Å. As an example, in the case when R = 4 Å the maximum curvature induced B ps is around 200 T, an order of magnitude larger than the hopping induced one (see Fig. 3d,e), in accordance with the finding of Rainis et al 45 . Both contributions to B ps are also dependent on the orientation of the fold within the graphene lattice, with the maximum B ps present in folds parallel to the zigzag direction and zero B ps for armchair 45,47 . For the double fold, the hopping induced B ps is in itself in the 200 Tesla range, due to the larger strains (Fig. 3f). A large collection of double or multiple folds may be necessary to create the large B ps values needed to observe the enhanced D' peak, such as in the region shown in red in Fig. 2b,c. We mention in passing that here we assume B ps is not homogeneous on the scale of the magnetic length, thus Landau quantization 48 due to strain is not expected. As an example, the magnetic length at 200 T is ≈1.7 nm.
Having quantified the B ps magnitude in folded, crumpled graphene, we turn our attention to numerically calculating the double resonant Raman processes in the presence of strain. Calculations are performed similarly to Venezuela et al 28 and Kürti et al 41 . For this we need to find the electron/hole -defect Hamiltonian H def . We model the B ps seen in crumpled graphene using a simple model, using a Gaussian deformation in graphene of the form: h · exp(− x 2 +y 2 b 2 ). For a Gaussian, the analytical form of the vector potential |A| ∝ h 2 /b 2 is well known 49 . By choosing the height h and width b of the bump to be 5 Å and 20 Å respectively, the B ps within the bump area reaches a maximum of 200 T, switching sign on the nm scale (see Fig. 4a), similarly to the values found within the LAMMPS calculations. In atomic units, the Hamiltonian describing the defect is H def = A(r)∇ + V (r), where A is the vector potential and we also include V , the scalar potential generated by the mechanical deformation 18 . In order to calculate the scattering matrix element k | H def | k ± q between two states with wave vectors k and k ± q we need to calculate A(±q) and V (±q) 28 : The calculated D, D' and 2D peaks for a Gaussian with h = 5 Å and b = 20 Å is shown in Fig. 4c. We reproduce I D /I D = 32 in accordance with the experimental spectrum of Fig. 2d. As a crosscheck to our calculations we also compute the D and D' intensity for a "hopping" defect, as implemented by Venezuela et al 28 , modeling a lattice defect in graphene. For the hopping defect, the D' peak intensity is 11% of the D intensity, as expected for lattice defects, such as vacancies 12 . Comparing the relative intensity with respect to the 2D peak, it is clear that the D' peak is showing a measurable intensity, as opposed to the D' peak generated by Coulomb scatterers 28 . The D' intensity is generated almost completely by the vector potential A, with the scalar potential giving only a slight (less then 10 −4 ) contribution. This can be easily explained if we consider the scattering matrix element between states | k | H def | k ± q | 2 = |H def (q)| 2 cos 2 (θ k,k±q /2), where θ k,k±q is the angle between the the initial and the scattered state 9 . For backscattering (θ k,k±q = π), if H def is of purely scalar character, the matrix element is zero. However, if H def has a vector potential component it can be nonzero. This is because A acts on the pseudospin of graphene, as evidenced by the pseudo-Zeeman effect 13 . Thus, it is able to change the phase of the charge carriers.
The vector potential A has a slight contribution to the D peak as well, as can be seen in Fig. 4c. This is due to the non zero scattering potential at large k values, around the K points. Due to the Fourier transformation in eq. 1, the contribution at large k increases as we make the Gaussian narrower. By decreasing b, but keeping the aspect ratio the same, we can see a marked increase in the scattering potential |A(k)| 2 at the K points. This contributes to a higher intensity of the D peak. This can be observed in Fig. 4d, where we plot the I D /I D ratio on a log scale, as a function of b and excitation energy.
Our experiments show a first example of resonant Raman scattering on potentials that are smooth on the atomic scale. In the lack of intervalley scattering, the pseudo-magnetic field created by the strain induced vector potential shifts the phase of the electron or hole wave function, in addition to the Berry phase. This lifts the restriction on backscattering, enabling an enhancement in the intravalley backscattering rate by orders of magnitude, evidenced by the enhancement of the D' peak in the Raman spectrum. The drastic increase in intravalley scattering, as a consequence of strain fluctuations, may be an important factor in lowering the mobility of certain CVD grown graphene samples, especially if wrinkles and folds are introduced during the transfer process. Furthermore, it is well known that defects such as grain boundaries, vacancies, etc. can have a specific I D /I D fingerprint 12 . Such defects also distort the graphene lattice around them 50 and this local strain field can be specific to the type of defect. Our results show that if we want to understand the origin of this Raman fingerprint, scattering on the defect induced local strain fields has to be taken into account.

Nanomaterials Research Group.
Author contributions PK carried out the experiments together with PN-I, the Raman calculations were carried out by GK under supervision from JK. PN-I carried out the molecular dynamics calculations. PN-I conceived and coordinated the project, together with LT and LPB. PN-I wrote the manuscript, with contributions from all authors.